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Fractional Calculus and Generalized Mittag-Leffler Type Functions

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Fractional Calculus and Generalized Mittag-Leffler Type Functions Fractional calculus and generalized Mittag-Leffler type functions Christian Lavault∗ Abstract In this paper, the generalized fractional integral operators of two generalized Mittag- Leffler type functions are investigated. The special cases of interest involve the generalized Fox–Wright function and the generalized M-series and K-function. In the next Section 2 we first recall some generalized fractional integral operators among the most widely used in fractional calculus. Section 3 is devoted to the definitions of M- series and K-function and their relations to special functions. In Sections 4 and 5, effective fractional calculus of the generalized M-series and the K-function is carried out. The last section briefly concludes and opens up new perspectives. The results established herein generalize recent properties of generalized Mittag-Leffler type functions using left- and right-sided generalized fractional differintegral operators. The note results also in important applications in physics and mathematical engineering. Keywords: Fox–Wright psi function; Generalized hypergeometric function; M-series and K-function; Mittag-Leffler type functions; Riemann–Liouville’s, Saigo’s and Saigo–Maeda’s generalized fractional calculus operators. 2010 Mathematics Subject Classification: 26A33, 33C05, 33C10, 33C20, 33C60, 44A15. Contents 1 Introduction and motivations 1 1.1 The Mittag-Leffler and generalized Mittag-Leffler type functions .............. 1 1.2 Integral representations of Mittag-Leffler type functions ................... 6 1.3 Higher transcendental functions . .............. 8 2 Fractional calculus of generalized M-L type functions 13 2.1 Riemann–Liouville fractional calculus . ................ 13 2.2 Saigo’s fractional differintegration operators . .................... 14 2.3 Saigo–Maeda’s fractional differintegration operators ..................... 16 arXiv:1703.01912v2 [math.CA] 21 Mar 2017 3 The generalized M-series and K-function 17 3.1 Relations to the Fox–Wright and M-L type functions . ................ 18 4 Fractional calculus of the M-series and the K-function 19 4.1 Left-sided generalized fractional integrations . ..................... 21 4.2 Right-sided generalized fractional integrations . ..................... 23 ∗LIPN, CNRS UMR 7030. E-mail: [email protected] 1 5 Fractional calculus involving F3 25 5.1 Left- and right-sided fractional integration of the M-series and the K-function . 25 5.2 Left- and right-sided fractional differentiation of the M-series and the K-function . 27 6 Conclusion & perspectives 28 Appendices 29 A Asymptotic expansion of M-L type functions (|z|→∞) ................... 29 B Determination of E−α,β(z) with negative value of the first parameter . 33 C Complex contour for the reciprocal gamma and the beta functions ............. 34 D Integral representation of the Gauß hypergeometric function................. 36 References 37 1 Introduction and motivations During the last two decades, the interest in Mittag-Leffler type functions has considerably developed. This is due to their vast potential of applications in applied sciences and engineer- ing, and their steadily increasing importance in physics researches. More precisely, deviations of physical phenomena having an exponential behavior may be governed by physical laws (expo- nential and power laws) with the help of generalized Mittag-Leffler type functions. For example, they appear especially important in research domains such as stochastic systems theory, dynam- ical systems theory, statistical distribution theory, disordered and chaotic systems, etc., with special emphasis placed on applications to fractional differential equations—although this topic is not addressed herein. Furthermore, geometric properties including starlikeness, convexity and close-to-convexity for the Mittag-Leffler type functions were also recently investigated, e.g. by Bansal and Prajapat in [4], Kilbas et al. [21] and Kiryakova [23, 24, 26]. This makes these func- tions directly and naturally amenable to fractional calculus techniques as studied by Gorenflo et al. [13], Kilbas et al. [17, 19, 21], Kiryakova [22], Kumar–Saxena [28], Saigo [37, 38, 39], Samko et al. [43], Saxena–Saigo [44], Sharma [46, 47], Srivastava et al. [49, 50], etc. 1.1 The Mittag-Leffler and generalized Mittag-Leffler type functions The one-parametric Mittag-Leffler function (M-L for short) Eα(z) was first introduced by the swedish mathematician G. M. Mittag-Leffler in five notes [31, 1903] [32, 1905] and also studied by Wiman [53, 1905]. It is a special function of z C which depends on the complex parameter α and is defined by the power series ∈ zn (1.1) E (z) := (α C). α Γ(αn + 1) ∈ nX≥0 One can see that the series (1.1) converges in the whole complex plane for all (α) > 0. For all (α) < 0, it diverges everywhere on C 0 and, for (α) = 0, its radius ofℜ convergence is R =eℜ π/2|ℑ(z)|. \{ } ℜ A first generalization of Eα(z) introduced by Wiman [53, 1905], and later studied by Agarwal et al. [1, 16, 1953], is the two-parametric M-L function of z C, defined by the series ∈ zn (1.2) E (z) := (α, β C; (α) > 0 (β) > 0). α,β Γ(αn + β) ∈ ℜ ℜ nX≥0 1 In the case when α and β are real positive, the series converges for all values of z C, while when α, β C, the conditions of convergence closely follow the ones for E (z) := E∈ (z). ∈ α,1 α Eα(z) and Eα,β(z) are entire functions of z C of order ρ = 1/ (α) and type σ = 1; in a sense, they are the simplest two entire functions∈ of this order (see,ℜ e.g., [13, §3.1]).1 The one- and two-parametric M-L function are fractional extensions of the basic functions E1( z) := ±z z ± E1,1( z)=e , E1,2(z) := (e 1)/z, E2(z) := E2,1(z) = cosh(√z), E2,2(z) = sinh(√z)/√z, etc. (see,± e.g., [2, §7.1], [10], [13,−Sec.1–4], [14], [17, 19, 22, 23, 53], and references therein). They also appear as solutions of fractional integro-differential equations, e.g. in [9, 21, 22, 24, 26]. Among the numerous generalizations of the M-L function let us point out the standard Wright function, defined by Wright in a bunch of papers (from 1933 to 1940) [54, 1934] by the series n 1 z −− (1.3) φ(α, β; z) := = ψ ; z (z, α, β C), Γ(αn + β) n! 0 1 (β,α) ∈ nX≥0 where 0ψ1( ; (β, α); z) is a special case of the generalized Fox–Wright psi function, which is given in Definition− 1.2, Eq. (1.24) of §1.3. The standard Wright function, which is very close to the two-parametric M-L function, may be rewritten also in terms of the Fox H-function introduced by Fox [11] (see Definition 1.4 in §1.3 and, e.g., [3, 17]). If α > 1, this series is absolutely convergent for all z C, while for α = 1, it is absolutely convergent− for z < 1, and for z =1 and (β) > 1 [21,∈Sec. 1.11]. − | | | | Forℜ α> −1, φ(α, β; z) is an entire function of z, wherefrom one can deduce that for α> 1, − 1 − the function has order ρ =1/(α +1) and type σ = (α + 1)α α+1 = αρ/ρ. δ Wright investigated the function φ(δ, ν + 1; z) := Jν (z), known as the Bessel-Wright func- tion (or the Wright generalized Bessel function)− and derived also the asymptotic behaviour of φ(α, β; z) at infinity by means of its integral representation in terms of a Mellin–Barnes contour integral (see, e.g., [22, App. E.ii],[21, 1.11]). Additionally, for z C ( , 0] and ν C, the functions φ(1, ν +1; z2/4) can be expressed also in terms of the Bessel∈ \ function−∞ of the first∈ kind, ± ∞ (z2/4)k z/2 J (z) := (z/2)ν ( 1)k = F ν + 1; z2/4 ν − k!Γ(ν + k + 1) Γ(ν + 1) 0 1 k=0 X 1 n Following Levin [29, Lect. 1], every entire function f(z) is represented by a power series f(z) = cnz which n≥0 n converges everywhere in the complex plane; the radius of convergence is infinite, which implies thatP lim |cn| = n→∞ 0. Any power series satisfying this criterion will represent an entire function. The global behaviour of an entire function of finite order is characterized by its order ρ and its type σ represented by the formulas p n log n 1 n ρ ρ = limsup and σ = lim sup n |cn| . n→∞ log (1/|cn|) ρe n→∞ Moreover, the asymptotic behaviour (|z|→∞) of an entire function is usuallyp studied via its restriction to rays in an angle θ1 ≤ | arg z| ≤ θ2 (cf. Phargmén–Lindelöf). The so-called indicator function of an entire function of order ρ is introduced as log f(reiθ ) h(θ) = limsup (θ1 ≤ θ ≤ θ2). rρ r→∞ σeρ n/ρ n Thus, the entire function f(z) = n z has order ρ and type σ. For instance, the M-L function ∞ (Aαz)n Γ(αn+1) , where A > 0 and αP > 0, is an entire function of order ρ = 1/α and type σ = A (by Stirling n=0 formula).P 2 and of the modified Bessel function ∞ (z2/4)k z/2 I (z) := (z/2)ν = F ν + 1; z2/4 , ν k!Γ(ν + k + 1) Γ(ν + 1) 0 1 − kX=0 as follows, φ(1, ν + 1; z2/4)=2z−νJ (z) and φ(1, ν + 1; z2/4)=2z−νI (z). − ν ν Both Bessel functions are analytic functions of z C, except for a branch point at z = 0 when ∈ ν is not an integer. The principal branches of Jν (z) and Iν (z) correspond to the principal value of (z/2)ν and is analytic in the z-plane cut along the interval ( , 0].
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